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## G = C2×C8.D14order 448 = 26·7

### Direct product of C2 and C8.D14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×C8.D14
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C2×C4○D28 — C2×C8.D14
 Lower central C7 — C14 — C28 — C2×C8.D14
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C8.D14
G = < a,b,c,d | a2=b8=1, c14=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c13 >

Subgroups: 1252 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×C8.C22, C56⋊C2, Dic28, C2×C56, C7×M4(2), C2×Dic14, C2×Dic14, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, C2×C56⋊C2, C2×Dic28, C8.D14, C14×M4(2), C22×Dic14, C2×C4○D28, C2×C8.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8.C22, C22×D4, D28, C22×D7, C2×C8.C22, C2×D28, C23×D7, C8.D14, C22×D28, C2×C8.D14

Smallest permutation representation of C2×C8.D14
On 224 points
Generators in S224
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)(121 193)(122 194)(123 195)(124 196)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 181)(138 182)(139 183)(140 184)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 197)(155 198)(156 199)(157 200)(158 201)(159 202)(160 203)(161 204)(162 205)(163 206)(164 207)(165 208)(166 209)(167 210)(168 211)
(1 46 8 39 15 32 22 53)(2 33 9 54 16 47 23 40)(3 48 10 41 17 34 24 55)(4 35 11 56 18 49 25 42)(5 50 12 43 19 36 26 29)(6 37 13 30 20 51 27 44)(7 52 14 45 21 38 28 31)(57 109 64 102 71 95 78 88)(58 96 65 89 72 110 79 103)(59 111 66 104 73 97 80 90)(60 98 67 91 74 112 81 105)(61 85 68 106 75 99 82 92)(62 100 69 93 76 86 83 107)(63 87 70 108 77 101 84 94)(113 150 134 157 127 164 120 143)(114 165 135 144 128 151 121 158)(115 152 136 159 129 166 122 145)(116 167 137 146 130 153 123 160)(117 154 138 161 131 168 124 147)(118 141 139 148 132 155 125 162)(119 156 140 163 133 142 126 149)(169 205 190 212 183 219 176 198)(170 220 191 199 184 206 177 213)(171 207 192 214 185 221 178 200)(172 222 193 201 186 208 179 215)(173 209 194 216 187 223 180 202)(174 224 195 203 188 210 181 217)(175 211 196 218 189 197 182 204)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 132 15 118)(2 117 16 131)(3 130 17 116)(4 115 18 129)(5 128 19 114)(6 113 20 127)(7 126 21 140)(8 139 22 125)(9 124 23 138)(10 137 24 123)(11 122 25 136)(12 135 26 121)(13 120 27 134)(14 133 28 119)(29 151 43 165)(30 164 44 150)(31 149 45 163)(32 162 46 148)(33 147 47 161)(34 160 48 146)(35 145 49 159)(36 158 50 144)(37 143 51 157)(38 156 52 142)(39 141 53 155)(40 154 54 168)(41 167 55 153)(42 152 56 166)(57 186 71 172)(58 171 72 185)(59 184 73 170)(60 169 74 183)(61 182 75 196)(62 195 76 181)(63 180 77 194)(64 193 78 179)(65 178 79 192)(66 191 80 177)(67 176 81 190)(68 189 82 175)(69 174 83 188)(70 187 84 173)(85 197 99 211)(86 210 100 224)(87 223 101 209)(88 208 102 222)(89 221 103 207)(90 206 104 220)(91 219 105 205)(92 204 106 218)(93 217 107 203)(94 202 108 216)(95 215 109 201)(96 200 110 214)(97 213 111 199)(98 198 112 212)

G:=sub<Sym(224)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,197)(155,198)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211), (1,46,8,39,15,32,22,53)(2,33,9,54,16,47,23,40)(3,48,10,41,17,34,24,55)(4,35,11,56,18,49,25,42)(5,50,12,43,19,36,26,29)(6,37,13,30,20,51,27,44)(7,52,14,45,21,38,28,31)(57,109,64,102,71,95,78,88)(58,96,65,89,72,110,79,103)(59,111,66,104,73,97,80,90)(60,98,67,91,74,112,81,105)(61,85,68,106,75,99,82,92)(62,100,69,93,76,86,83,107)(63,87,70,108,77,101,84,94)(113,150,134,157,127,164,120,143)(114,165,135,144,128,151,121,158)(115,152,136,159,129,166,122,145)(116,167,137,146,130,153,123,160)(117,154,138,161,131,168,124,147)(118,141,139,148,132,155,125,162)(119,156,140,163,133,142,126,149)(169,205,190,212,183,219,176,198)(170,220,191,199,184,206,177,213)(171,207,192,214,185,221,178,200)(172,222,193,201,186,208,179,215)(173,209,194,216,187,223,180,202)(174,224,195,203,188,210,181,217)(175,211,196,218,189,197,182,204), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,132,15,118)(2,117,16,131)(3,130,17,116)(4,115,18,129)(5,128,19,114)(6,113,20,127)(7,126,21,140)(8,139,22,125)(9,124,23,138)(10,137,24,123)(11,122,25,136)(12,135,26,121)(13,120,27,134)(14,133,28,119)(29,151,43,165)(30,164,44,150)(31,149,45,163)(32,162,46,148)(33,147,47,161)(34,160,48,146)(35,145,49,159)(36,158,50,144)(37,143,51,157)(38,156,52,142)(39,141,53,155)(40,154,54,168)(41,167,55,153)(42,152,56,166)(57,186,71,172)(58,171,72,185)(59,184,73,170)(60,169,74,183)(61,182,75,196)(62,195,76,181)(63,180,77,194)(64,193,78,179)(65,178,79,192)(66,191,80,177)(67,176,81,190)(68,189,82,175)(69,174,83,188)(70,187,84,173)(85,197,99,211)(86,210,100,224)(87,223,101,209)(88,208,102,222)(89,221,103,207)(90,206,104,220)(91,219,105,205)(92,204,106,218)(93,217,107,203)(94,202,108,216)(95,215,109,201)(96,200,110,214)(97,213,111,199)(98,198,112,212)>;

G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,197)(155,198)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211), (1,46,8,39,15,32,22,53)(2,33,9,54,16,47,23,40)(3,48,10,41,17,34,24,55)(4,35,11,56,18,49,25,42)(5,50,12,43,19,36,26,29)(6,37,13,30,20,51,27,44)(7,52,14,45,21,38,28,31)(57,109,64,102,71,95,78,88)(58,96,65,89,72,110,79,103)(59,111,66,104,73,97,80,90)(60,98,67,91,74,112,81,105)(61,85,68,106,75,99,82,92)(62,100,69,93,76,86,83,107)(63,87,70,108,77,101,84,94)(113,150,134,157,127,164,120,143)(114,165,135,144,128,151,121,158)(115,152,136,159,129,166,122,145)(116,167,137,146,130,153,123,160)(117,154,138,161,131,168,124,147)(118,141,139,148,132,155,125,162)(119,156,140,163,133,142,126,149)(169,205,190,212,183,219,176,198)(170,220,191,199,184,206,177,213)(171,207,192,214,185,221,178,200)(172,222,193,201,186,208,179,215)(173,209,194,216,187,223,180,202)(174,224,195,203,188,210,181,217)(175,211,196,218,189,197,182,204), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,132,15,118)(2,117,16,131)(3,130,17,116)(4,115,18,129)(5,128,19,114)(6,113,20,127)(7,126,21,140)(8,139,22,125)(9,124,23,138)(10,137,24,123)(11,122,25,136)(12,135,26,121)(13,120,27,134)(14,133,28,119)(29,151,43,165)(30,164,44,150)(31,149,45,163)(32,162,46,148)(33,147,47,161)(34,160,48,146)(35,145,49,159)(36,158,50,144)(37,143,51,157)(38,156,52,142)(39,141,53,155)(40,154,54,168)(41,167,55,153)(42,152,56,166)(57,186,71,172)(58,171,72,185)(59,184,73,170)(60,169,74,183)(61,182,75,196)(62,195,76,181)(63,180,77,194)(64,193,78,179)(65,178,79,192)(66,191,80,177)(67,176,81,190)(68,189,82,175)(69,174,83,188)(70,187,84,173)(85,197,99,211)(86,210,100,224)(87,223,101,209)(88,208,102,222)(89,221,103,207)(90,206,104,220)(91,219,105,205)(92,204,106,218)(93,217,107,203)(94,202,108,216)(95,215,109,201)(96,200,110,214)(97,213,111,199)(98,198,112,212) );

G=PermutationGroup([[(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,81),(16,82),(17,83),(18,84),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,100),(56,101),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192),(121,193),(122,194),(123,195),(124,196),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176),(133,177),(134,178),(135,179),(136,180),(137,181),(138,182),(139,183),(140,184),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,218),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,197),(155,198),(156,199),(157,200),(158,201),(159,202),(160,203),(161,204),(162,205),(163,206),(164,207),(165,208),(166,209),(167,210),(168,211)], [(1,46,8,39,15,32,22,53),(2,33,9,54,16,47,23,40),(3,48,10,41,17,34,24,55),(4,35,11,56,18,49,25,42),(5,50,12,43,19,36,26,29),(6,37,13,30,20,51,27,44),(7,52,14,45,21,38,28,31),(57,109,64,102,71,95,78,88),(58,96,65,89,72,110,79,103),(59,111,66,104,73,97,80,90),(60,98,67,91,74,112,81,105),(61,85,68,106,75,99,82,92),(62,100,69,93,76,86,83,107),(63,87,70,108,77,101,84,94),(113,150,134,157,127,164,120,143),(114,165,135,144,128,151,121,158),(115,152,136,159,129,166,122,145),(116,167,137,146,130,153,123,160),(117,154,138,161,131,168,124,147),(118,141,139,148,132,155,125,162),(119,156,140,163,133,142,126,149),(169,205,190,212,183,219,176,198),(170,220,191,199,184,206,177,213),(171,207,192,214,185,221,178,200),(172,222,193,201,186,208,179,215),(173,209,194,216,187,223,180,202),(174,224,195,203,188,210,181,217),(175,211,196,218,189,197,182,204)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,132,15,118),(2,117,16,131),(3,130,17,116),(4,115,18,129),(5,128,19,114),(6,113,20,127),(7,126,21,140),(8,139,22,125),(9,124,23,138),(10,137,24,123),(11,122,25,136),(12,135,26,121),(13,120,27,134),(14,133,28,119),(29,151,43,165),(30,164,44,150),(31,149,45,163),(32,162,46,148),(33,147,47,161),(34,160,48,146),(35,145,49,159),(36,158,50,144),(37,143,51,157),(38,156,52,142),(39,141,53,155),(40,154,54,168),(41,167,55,153),(42,152,56,166),(57,186,71,172),(58,171,72,185),(59,184,73,170),(60,169,74,183),(61,182,75,196),(62,195,76,181),(63,180,77,194),(64,193,78,179),(65,178,79,192),(66,191,80,177),(67,176,81,190),(68,189,82,175),(69,174,83,188),(70,187,84,173),(85,197,99,211),(86,210,100,224),(87,223,101,209),(88,208,102,222),(89,221,103,207),(90,206,104,220),(91,219,105,205),(92,204,106,218),(93,217,107,203),(94,202,108,216),(95,215,109,201),(96,200,110,214),(97,213,111,199),(98,198,112,212)]])

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 28 28 2 2 2 2 28 ··· 28 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 D28 D28 C8.C22 C8.D14 kernel C2×C8.D14 C2×C56⋊C2 C2×Dic28 C8.D14 C14×M4(2) C22×Dic14 C2×C4○D28 C2×C28 C22×C14 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C14 C2 # reps 1 2 2 8 1 1 1 3 1 3 6 12 3 18 6 2 12

Matrix representation of C2×C8.D14 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 104 46 0 0 0 0 67 9 0 0 0 0 0 0 112 0 69 0 0 0 0 37 0 36 0 0 0 1 1 0 0 0 1 31 0 76
,
 34 25 0 0 0 0 88 88 0 0 0 0 0 0 112 44 0 0 0 0 77 1 0 0 0 0 36 36 1 36 0 0 36 67 69 112
,
 18 42 0 0 0 0 108 95 0 0 0 0 0 0 27 66 58 52 0 0 36 95 2 68 0 0 0 29 86 81 0 0 75 104 67 18

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[104,67,0,0,0,0,46,9,0,0,0,0,0,0,112,0,0,1,0,0,0,37,1,31,0,0,69,0,1,0,0,0,0,36,0,76],[34,88,0,0,0,0,25,88,0,0,0,0,0,0,112,77,36,36,0,0,44,1,36,67,0,0,0,0,1,69,0,0,0,0,36,112],[18,108,0,0,0,0,42,95,0,0,0,0,0,0,27,36,0,75,0,0,66,95,29,104,0,0,58,2,86,67,0,0,52,68,81,18] >;

C2×C8.D14 in GAP, Magma, Sage, TeX

C_2\times C_8.D_{14}
% in TeX

G:=Group("C2xC8.D14");
// GroupNames label

G:=SmallGroup(448,1200);
// by ID

G=gap.SmallGroup(448,1200);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,297,80,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^14=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations

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